Specifically, we call this system the Walsh-Paley one
if \(m_k= 2, \ \text{for all} \ k\in \mathbb {N}.\)  In this case we have dyadic group \(G_2=\prod _{j=0}^{\infty }Z_{2},\)  which is called the Walsh group and Vilenkin system coincides Walsh functions defined by (for details see e.g. [1]} and [2]})
\(w_{n}(x):=\underset{k=0}{\overset{\infty }{\prod }}r_{k}^{n_{k}}\left(x\right) =r_{\left|n\right|}\left( x\right) \left( -1\right) ^{\underset{k=0}{\overset{\left|n\right|-1}{\sum }}n_{k}x_{k}}\text{\qquad }\left( n\in \mathbb {N}\right),\) 
